Computed rate coefficients and product yields for cC[subscript 5] H[subscript 5] + CH[subscript 3] [right
arrow] products
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Citation
S. Sharma and W.H. Green, “Computed rate coefficients and
product yields for c-C[subscript 5] H[subscript 5] + CH[subscript
3] [right arrow] products,” The Journal of Physical Chemistry A,
vol. 113, 2009, pp. 8871-8882.
As Published
http://dx.doi.org/10.1021/jp900679t
Publisher
American Chemical Society
Version
Author's final manuscript
Accessed
Fri May 27 00:10:55 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/49489
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Computed rate coefficients and product yields for
c-C5H5 + CH3 → products
Sandeep Sharma, William H. Green
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
June 15, 2009
Abstract
Using quantum chemical methods we have explored the region of the C6 H8
potential energy surface that is relevant in predicting the rate coefficients of various
wells and major product channels following the reaction c-C5 H5 +CH3 . Variational
transition state theory is used to calculate the high-pressure-limit rate coefficient
for all the barrierless reactions. RRKM theory and the master equation is used to
calculate the pressure dependent rate coefficients for 13 reactions. The calculated
results are compared with the limited experimental data available in literature and
the agreement between the two is quite good. All the rate coefficients calculated
in this work are tabulated and can be used in building detailed chemical kinetic
models.
1
Introduction
c-C5 H5 +CH3 is an important pathway towards benzene formation in combustion and
pyrolysis. 1 Although the potential energy surface was computed in the 1990’s by Melius
et al. 2 and Moskaleva et al. 3 more precise calculations are now possible. Even with an
accurate PES the calculation of the pressure dependent gas-phase rate coefficients is a
challenging task. As can be seen in the potential energy surface in Figure 1, several
loose transition states and chemically activated product channels are important at high
temperatures.
Recently there have been considerable advances in the methodology for rate coefficient
calculations for loose transition states 4–7 . In addition to these advances there has been
significant progress in accurate calculation of the pressure dependent rate coefficients for
complicated potential energy surfaces with multiple wells and multiple product channels. 8–13 The method is based on the master equation formulation of the problem. These
detailed equations are then appropriately coarse-grained to calculate the phenomenological rate coefficients.
In this paper we use both these methods to calculate the pressure dependent rate
coefficients for the title reaction over a range of temperatures and pressures. These
pressure dependent rate coefficients, the high pressure limit rate coefficients and the
thermochemistry of all the reactions and species are provided for use in a detailed chemical
kinetic models.
1
2
Theoretical Methodology
Figure 1 shows the potential energy diagram for the formation of Fulvene and various
C6 H8 and C6 H7 species by the title reaction. In addition to the product channels labeled
in the figure we have also calculated the energy of other high energy product channels and
tabulated their energy relative to c-C5 H5 +CH3 in Table 1. For this study we have included
only the 5 product channels shown in Figure 1. R5 channel has not been included even
though it is lower energy channel than R4 because there is a higher barrier towards the
formation of R5 and when we performed calculations by including this product channel its
rate coefficient was lower than the R4 product channel by about an order of magnitude.
The nomenclature used in Figure 1 and Table 1 is explained by Figure 2.
CH2*
+ H*
40
C
R4+H
D
CH2*
+ H*
20
C*
Energy (kcal/mole)
E
0
c-C5H5
+ CH3
R3+H
R1 + H
R2+H
CH2*
+ H*
-20
+ H2
-40
A
B
Fulvene+H2
-60
C5H5CH3-5
-80
C5H5CH3-1
C5H5CH3-2
Figure 1: Potential Energy diagram at CBS-QB3 level of theory for reaction of c-C5 H5
with CH3 calculated at 0 K.
2
+ H*
Molecule
Fulvene+H2
R2+H
R1+H
R3+H
R5+H2
R4+H
C5 H5 -3+CH3
C5 H5 -2+CH3
R11+H
R9+H
R8+H
R12+H
R10+H
R7+H
R6+H
Energy
(kcal/mole)
-47.13
7.20
7.34
13.00
20.73
31.10
33.07
33.59
40.77
41.14
41.55
41.71
41.81
44.07
44.76
Table 1: The various possible product channels and their energy relative to the entrance
channel c-C5 H5 +CH3 arranged in ascending order. All the energies are calculated using
CBS-QB3 method and are at 0 K.
2.1
2.1.1
Quantum Chemistry
Single-Reference Methods
All single reference quantum chemical calculations described in this section are performed
by Gaussian03 quantum chemistry package. 14 The potential energy diagram in Figure 1
was generated by performing calculations at CBS-QB3 level of theory. 15–17 The method
makes use of the fact that geometry is not very sensitive to the level of theory employed
and so the geometry and frequencies are calculated at B3LYP/CBSB7 level of theory.
This geometry is then used to perform higher level quantum calculations at CCSD(T)/631g+(d) and MP4SDQ/CBSB4. The energies at these level are used with the final step
which incorporates some empirical correction factors to estimate energies of CCSD(T)
method with an infinite basis set. This method is widely used and should give energies
accurate to within 2 kcal/mole. 17
Ab-initio calculations for all the stable molecules, radicals and the tight transition
states were also performed using G2 method, 18 another method which should provide
energies with an accuracy in the range of 2 kcal/mole. 19 G2 method is based on a similar
idea as CBS-QB3 method, where the optimized geometry is calculated at MP2/6-31G(d)
level of theory and frequencies are calculated at HF/6-31G(d) level of theory. Subsequently higher level methods including MP2/6-311+G(3df,2p), QCISD(T)/6-311G(d,p)
and MP4/6-311+G(d,p) are used to calculate correlation energies. Finally an empirical
correction is added to correct for correlation ignored by QCISD(T) method and also to
extrapolate to the infinite basis set limit. These two methods differ substantially in the
type of extrapolation scheme they employ and give a set of values which can be compared
3
to each other to check anomalously high differences in energies. For the transition states
(a) c-C5 H5
(b) C5 H5 CH3 -5
(c) C5 H5 CH3 -1
(d) C5 H5 CH3 -2
(e) R1
(f) R2
(g) R3
(h) R4
(i) R5
(j) R6
(k) R7
(l) R8
(m) R9
(n) R10
(o) R11
(s) A
(p) R12
(t) B
(q) C5 H5 -2
(u) C
(r) C5 H5 -3
(v) D
(w) E
Figure 2: Names and structure of molecules and transition states
the negative frequency was visualized to validate the correctness of transition state.
2.1.2
Multireference Methods
In the potential energy surface shown in Figure 1 there are 7 transition states without an
energy barrier. To compute the rate coefficients for these reactions we use microcanonical
4
variational transition state theory, which requires an accurate energy prediction of the
potential energy along the bond breaking coordinate. Conventional single reference abintio methods fail to accurately predict the potential during bond breaking after the bond
length increases beyond a certain length. For this reason we use CASPT2 method 20,21
which is a multireference counterpart of the MP2 method, as implemented in MOLPRO. 22
The key problem in these methods is the selection of an appropriate active space. In
general the active space should include all the orbitals taking part in the bond breaking
plus the π and π ∗ orbitals in the ring. First MCSCF calculations are performed with a
given basis set and active space. The orbitals of the MCSCF calculations are used to then
perform the CASPT2 calculations. The active space to include in CASPT2 calculations
can be judged by looking at the occupation number of the orbitals.
Orbital
19
20
21
22
23
24
25
26
2.3
1.990
1.947
1.906
1.791
0.220
0.070
0.065
0.010
2.5
1.990
1.944
1.904
1.689
0.323
0.072
0.068
0.010
C–C bond length (A)
2.7
2.9
3.2
3.6
4.0
1.990 1.990 1.990 1.991 1.991
1.941 1.939 1.938 1.937 1.937
1.903 1.902 1.901 1.901 1.901
1.573 1.459 1.315 1.183 1.105
0.439 0.554 0.698 0.829 0.907
0.073 0.074 0.074 0.075 0.076
0.070 0.072 0.074 0.074 0.075
0.010 0.010 0.010 0.009 0.009
5.0
1.991
1.937
1.901
1.052
0.959
0.076
0.075
0.009
6.0
1.991
1.937
1.901
1.026
0.985
0.076
0.075
0.009
Table 2: The occupation number of different orbitals in the active space for various C-C
bond distances, using cc-pvdz basis.
We first calculate the potential for C-C bond fission in methylcyclopentadiene to form
methyl and cyclopentadienyl radials. Table 2 shows the occupation numbers of different
orbitals in the active space, when the active space consists of 8 orbitals and 8 electrons.
The criteria we have used in our work is, if an orbital has an occupation number greater
than 1.97 or less than 0.03 then we do not include it in the active space. The table
shows that at all bond lengths the 19th and 26th orbitals don’t need to be included in the
active space based on our criteria. Figure 3 confirms that including electrons and orbitals
beyond the required 6 doesn’t add to the accuracy of the method. The potential energy
calculated using 6e6o active space and 8e8o active space is the same, but the shape of the
potential changes significantly if we make the active space smaller to 4e4o as is shown in
Figure 3.
On the other hand the same figure shows that the inclusion of a larger basis set cc-pvtz
as opposed to cc-pvdz makes the potential slightly (∼ 1 kcal/mole) more attractive. This
same effect was seen in some of the earlier papers by Klippenstein and Harding. 5,6 The
cc-pvdz surface closely matches the cc-pvtz surface suggesting the basis set expansion has
sufficiently converged by cc-pvtz.
5
2
0
Energy (kcal/mole)
-2
-4
-6
-8
-10
-12
-14
CASPT2(8e,8o)/cc-pvdz
CASPT2(6e,6o)/cc-pvdz
CASPT2(4e,4o)/cc-pvdz
CASPT2(6e,6o)/cc-pvtz
-16
-18
-20
2
2.4
2.8
3.2
3.6
4
4.4
4.8
5.2
5.6
6
C-C bond distance (A)
Figure 3: Comparison of Potential curves using CASPT2 method with different active
spaces and basis sets. Increasing the active space beyond 6e,6o does not change the result
significantly. The basis set of cc-pvtz makes the potential more attractive compared to
cc-pvdz.
2.2
Statistical Mechanics and Rate Calculations
For most of the molecules and the tight transition states we have used rigid rotor harmonic
oscillator approximation with corrections added for internal rotors. The frequencies calculated using B3LYP/CBSB7 theory were scaled by a factor of 0.99 as recommended
by Scott et al. 23 In molecules where there is a single bond between sp3 and sp2 carbon
atoms we assumed that the rotation about that bond was barrierless so it was treated
as a free rotor (e.g. the out of ring carbon carbon single bond in C5 H5 CH3 -1 has a rotational barrier of only 1.4 kcal/mole). The barriers for other internal hindered rotors were
calculated by a relaxed scan at B3LYP/CBSB7 level.
P The scanned potential was fit to
a Fourier series in the torsional angle φ; V (φ) =
(Am cos(mφ) + Bm sin(mφ)) where
m went from 0 to 5. The reduced moment of inertia based on the equilibrium geometry
was calculated using the formulas described by Pitzer. 24–26 In this paper we have used
the reduced moment of inertia I (2,3) , 27 and the one dimensional Schroedinger equation
was solved numerically. For cyclopentadienyl an additional moment of inertia for pseudo
rotation was used as described in Section 3.1. For the calculations of heats of formation
of stable molecules, bond additivity corrections (BAC) as recommended by Petersson 28
were also included. This BAC correction was not included in the calculation of the rate
coefficients because reliable BAC values are not available for transition states and including BAC only in reactants can cause systematic errors in barrier heights. For tight
6
transition states canonical transition state theory was used and tunneling was included
by simple Wigner correction. 29
2.2.1
Density of State Calculations in Variflex
In the calculations involving loose transition states and pressure dependent calculations
Variflex was used to calculate density of states and subsequently the rate coefficients. The
details of the calculations have been described in Miller et al. 9 For all stable molecules and
tight transition states the rigid rotor harmonic oscillator approximation was used. The
classical density of states corresponding to the hindered torsional motions was calculated
as a phase space integral. An estimate of the quantum density of states was obtained
using Pitzer-Gwinn 24 approximation shown in Equation 1, where ρtq (E) and ρtcl (E) are
ho
quantum mechanical and classical density of states for torsion and ρho
q (E) and ρcl (E) are
the quantum mechanical and classical density of states obtained by treating the torsion
as a harmonic oscillator.
ρho
q (E)
ρtq (E) = ρtcl (E) ho
(1)
ρcl (E)
The difference between this estimate and the density of states for hindered rotors computed by solving the 1-d Schroedinger equation is not large for our system. The total
sum of states N(E) was calculated by convoluting this hindered rotor density of state
with the density of states of the other vibrational modes. The rotational states were
calculated by assuming that in molecules at least two of the three principal moments of
inertia were equal i.e. asymmetric tops were approximated as symmetric tops. For any
given value of rotational quantum number J total number of states including vibrational
and rotational states is calculated to obtain N(E, J) by a method described by Miller et
al. 30
2.3
Variational Transition State Rate Calculation
We have used Variflex to calculate the rate coefficients for the loose transition states. A
brief description of the method is presented below. The E,J resolved microscopic rate
coefficient due to RRKM 31,32 theory is given in Equation 2.
k(E, J) =
N(E, J)
hρ(E, J)
(2)
where the numerator is the sum of states of the transition state to energy E with
angular momentum J and the denominator is the density of states of the reactants at the
specified energy and angular momentum. In variational transition state theory a dividing
hypersurface which separates the reactants from the products is picked which minimizes
the numerator in Equation 2.
Here we use the assumption 33–35 whereby the internal modes of the transition states
are divided into 1 reaction coordinate, 5 transitional modes (for two non-linear fragments) which involve relative rotations of the fragments and the remaining “conserved
modes” which are vibrations of the separated fragments and are assumed to not change
substantially along the reaction coordinate. The density of states of the conserved modes
are calculated by applying the traditional rigid rotor harmonic oscillator approximation
7
in which the frequencies of the separated fragments are used. The transitional modes
are usually low frequency modes and the number of states of these modes with angular
momentum Jℏ and energy less than or equal to E at the dividing surface s = s0 can be
calculated using the semi-classical phase space integral shown in Equation 3. 36
Z
ℏ
Ntr (E, J, s0 ) = n δ(s − s0 )δ(E − H(τ, ps , s))δ(Jℏ − JT (τ, ps , s))Θ(ṡ)ṡ dτ dsdps (3)
h
In this equation, s is the reaction coordinate and ps is its conjugate momentum, τ has n
coordinates, conjugate momenta pair which include the transitional coordinates, conjugate momenta pairs and an additional three pairs corresponding to external rotation of
the whole molecule (for 2 nonlinear fragments n = 8), H(τ ) is the classical Hamiltonian
of the transitional modes, JT is the magnitude of the classical angular momentum and
Θ(ṡ) is the step function of ṡ which is 0 for ṡ < 0 and 1 otherwise.
The reaction coordinate (s) can be picked in many ways, the easiest two being distance
between the reacting atoms and the distance between the center of masses of the two
fragments. For a rigorous calculations not only the transition state along the reaction
coordinate is varied but also the definition of the reaction coordinate itself is varied to
minimize the TST reaction rate. In our study we have decided to simplify the analysis by
picking the definition of the reaction coordinate as the distance between the two bonding
atoms. It has been previously pointed out 4 that the change of pivot point location within
the reacting fragments can cause the computed rate of the reaction to decrease by 5%.
The integral in Equation 3 contains 3 delta functions which are separately integrated
out before using a Monte Carlo method for calculating the integral. The method has
been described in detail by Klippenstein. 36 The sum of states of transitional modes is
convoluted with the density of states of the conserved modes to get the total sum of
states of a transition state.
The Hamiltonian H(τ ) is the sum of kinetic energy and the potential energy. The
potential energy is written as a sum of changes in potential (Equation 4)
1. due to the change in the separation between the two fragments at a given interfragment orientation and
2. due to change in the inter-fragment orientation at a given separation.
V (s, τ ) ≈ V1 (s, τf ixed ) + V2 (s, τ ) − V2 (s, τf ixed )
(4)
Usually calculation of the first part of this potential (V1 ) requires a high level of
accuracy because inter-fragment separation represents bond-breaking or bond-forming,
while the second part of this potential (V2 ) is relatively easy to predict so a lower level
method can be used. This same principle was used by Petersson 37 for development of
the IRC-MAX calculations and by Klippenstein and Harding in their papers. 4,5
To test this assumption we have calculated the high pressure rate coefficient for the entrance channel c-C5 H5 +CH3 using CASPT2(6e,6o)/cc-pvdz method and CASPT2(6e,6o)/ccpvdz+CASPT2(2e,2o)/cc-pvdz method. CASPT2(6e,6o)/cc-pvdz represents the method
in which the potential used in the Monte-Carlo method is calculated at CASPT2(6e,6o)/ccpvdz level of theory, on the other hand CASPT2(6e,6o)/cc-pvdz+CASPT2(2e,2o)/ccpvdz represents a method in which V1 is calculated using CASPT2(6e,6o)/cc-pvdz level of
8
theory and the orientation dependence of the potential V2 is calculated using CASPT2(2e,2o)/ccpvdz. The results are plotted as shown in Figure 4. The maximum difference between the
rate coefficients computed using the two methods is 35%. For our work here we deem this
as an acceptable level of accuracy given the fact that we get about an order of magnitude
saving in computation time.
k (cm3/molecule/s)
1e-09
1e-10
1e-11
0
400
800
1200
1600
2000
2400
Temperature (K)
Figure 4: High pressure limit rate coefficients for reaction c-C5 H5 +CH3 → C5 H5 CH3 -5
predicted by CASPT2(6e,60)/cc-pvdz+CASPT2(2e,2o)/cc-pvdz method (solid line) and
CASPT2(6e,6o)/cc-pvdz method (dashed line).
Rate coefficients for all the loose transition states presented in this paper were calculated using CASPT2(6e,60)/cc-pvtz+CASPT2(2e,2o)/cc-pvdz. For each transition state
we have picked 9 interatomic separations ranging from 2.3 Å to 6.0 Å. For each separation
we carried out 1000 CASPT2(2e,2o)/cc-pvdz calculations to calculate the sum of states
of the transitional modes using Monte Carlo method.
Our separation of conserved and the transitional modes neglects the change in geometry of the reacting fragments as they move closer to each other. It has been pointed out
in an earlier paper by Harding et al. 4 that this approximation can cause the transition
state position to decrease to very low inter-atomic separation at high E’s and J’s. In
our work we have noticed the same trend and have restricted the loose transition state
position to have interfragment distance > 2.3 Å .
The J-averaged microcanonical rate coefficient k(E) was computed by
P
1 J N ‡ (E, J)(2J + 1)∆J
P
(5)
k(E) =
h
J ρ(E, J)(2J + 1)∆J
where ∆J is the discretization used for J, N ‡ (E, J) and ρ(E, J) are the sum of states
of transition state and density of states of the reactant respectively; the canonical rate
9
Bath Gas
Ar
Kr
N2
< ∆Ed >
100
350
125
Table 3: Energy transfer parameters used in the exponential down model for bath gas
Ar (see Section 3.4), Kr 40 and N2 . 11 All values have units of cm−1 .
constant k(T ) was computed by
P P
1 E J N ‡ (E, J)(2J + 1)e−E/kb T ∆J∆E
P P
k(T ) =
−E/kb T ∆J∆E
h
E
J ρ(E, J)(2J + 1)e
(6)
where ∆E is the discretization used for E. In this work we have used ∆E = 75 cm−1
and ∆J = 25.
2.4
Master Equation and Phenomenological Rate Constant
We have calculated the pressure dependent rate coefficients for all the reactions in the
PES shown in Figure 1 using Variflex. To calculate these pressure dependent rate coefficients Variflex solves the 1-D master chemical equation and uses the energy resolved
microcanonical rate coefficients (k(E)) as calculated in Equation 5. For all the calculations the energy transfer probability for each collision assumes the exponential down
model, with the energy transfer parameters for each bath gas used in this study provided
in Table 2.4. The details of the formulation of the master equation and methodology of
solution used in Variflex is described in detail elsewhere. 11–13,38,39
3
3.1
Results and Discussions
Thermochemistry
We have performed CBS-QB3 and G2 calculations on each of the molecules in Figure 1.
Figure 5 shows the difference between the atomization energy of different molecules and
transition states calculated using G2 and CBS-QB3 method. The atomization energies
predicted by G2 are systematically higher than the CBS-QB3 values. It has been noted
earlier that the G2 method is deficient when it comes to predicting the heat of formation
for unsaturated cyclic molecules. 41 To test this hypothesis we have performed calculations at G3 41 level for the molecules which showed considerable discrepancy between the
G2 and CBS-QB3 predictions. These molecules include c-C5 H5 , R1 and R2. The differences between the atomization energies between G3 and CBS-QB3 values for these three
molecules are -0.29, -1.01 and 0.50 kcal/mole respectively, while the difference between
G2 and G3 are much higher. These small differences in G3 and CBS-QB3 atomization
energies suggest that CBS-QB3 values are more accurate than G2 values for the present
system and we therefore used the energies calculated by CBS-QB3 method. The predicted
thermochemical properties for all the molecules included in this study are tabulated in
10
Table 5. Also the thermochemical values calculated by Melius et al. 2 using the BAC-MP4
method are compared to the ones calculated here using the CBS-QB3 method in Table 6.
c-C5 H5
There have been several studies of the Jahn-Teller effect on the energy levels of c-C5 H5 ,
which should allow accurate computation of its thermochemistry. 42–44 These studies recommend the usual rigid rotor harmonic oscillator approach for the treatment of the non
Jahn-Teller active vibrational modes and a more rigorous treatment of Jahn-Teller vibrational modes. The treatment of the Jahn-Teller active modes is complicated by the
coupling of the vibrational and electronic states due to the near degenerate electronic
state of the molecule. In the rest of the section we give a brief description of this coupling and how its effect can be taken into account to calculate the vibronic states. For a
more detailed derivation the reader can refer to one of the many good treatments available. 42,45,46 Our purpose here is to give enough detail so that the reader can appreciate
what approximation we have used in writing our code which calculates the Jahn-Teller
vibronic states.
We expand the vibronic states of the molecule as a summation given in Equation 7
where |Λ(qe )i are the eigenstates of the electronic Hamiltonian defined in Equation 9.
In D5h symmetry of cyclopentadienyl the ground state is two-fold degenerate and the
next excited state is energetically removed from them. Then due to Born-Oppenheimer
approximations the summation over all electronic states is truncated to just the ones
over the two degenerate ground states shown in Equation 8. In subsequent equations
in this section we use the notation that qe and pe are the coordinates and conjugate
momentum of the electrons and Qn and Pn are the normal coordinates of the nuclei and
their conjugate momenta. The origin of the nuclear coordinates is chosen so that all the
normal coordinates Qi in the vector Qn are 0 at the lowest energy D5h geometry and at
this geometry Qn is represented by Q0n .
|Ψi =
X
|Λ(qe )i|ξ Λ (Qn )i
(7)
n,Λ
1
|Ψi = |Λ (qe )i|ξ 1 (Qn )i + |Λ−1(qe )i|ξ −1(Qn )i
Hel (qe , Q0n )|Λ(qe )i =
X P2
e
e
2
!
+ V (qe , Q0n ) |Λ(qe )i = Eel |Λ(qe )i
(8)
(9)
The next step is to expand the nuclear component of the wavefunction |ξ(Qn )i as
a linear combination of products of harmonic oscillator basis functions centered on the
lowest-energy D5h geometry as shown in Equation 10. In Equation 10 |νji i is ket describing the wave function of the ith non Jahn-Teller vibrational mode with quantum number
νji and |νjdi , ljdii describes the wave function of the ith degenerate Jahn-Teller mode in
the state with quantum numbers νjdi and ljdi .
|ξ 1 (Qn )i =
X
A1j1,..,jd1,..,jdp|νj1 i|νj2i...|νj3N −6−2p i|νjd1 , ljd1 i|νjd2, ljd2 i...|νjdp , ljdp i
νj1 ,..,νjdp ,ljdp
(10)
11
Stable Molecules
∆ Energy (kcal/mole)
4
3
2
1
0
-1
4
R
3
R
2
R
1
R
e
en 2
lv
Fu H 3
C
H 5 -1
C5 H3
C
H 5 -5
C5 H3
C
H5
C5
H3
C
H5
C5
c-
Transition States
∆ Energy (kcal/mole)
2
1
0
-1
-2
A
B
C
E
Figure 5: Difference between the atomization energy using G2 method and CBS-QB3
method. In the second figure all the transition states are as labeled in the PES. We
could not obtain saddle point for transition state D using the G2 method and hence the
comparison is not given here.
12
The full electronic plus nuclear Hamiltonian is written and simplified to a form given
in Equation 11. In this equation the first and second derivatives have been calculated at
the lowest energy D5h geometry.
H(q, Q) =
X P2
!
n
+ Hel (Q0n ) + ∆V (qe , Qn )
n
=
X P2
n
=
X P2
n
e
+
+ V (qe , Qn )
2
2m
e
e
2
X P2
X ∂2V
X ∂V
Qi +
Q2
+ Hel (Q0n ) +
2 i
2
∂Q
∂Q
i
i
i
i
n
n
!
!
(11)
We can write the Hamiltonian operator as a matrix using the basis set defined in
Equation 8. The resulting Hamiltonian matrix is diagonalized to calculate the vibronic
energy levels. For a full derivation of the matrix elements for a Jahn-Teller molecule we
refer the reader to comprehensive review written by Barckholtz and Miller. 42
∂V
|Λ∓ (qe )i, where Qi represents a norIn Equation 11 the non-zero value of hΛ± (qe )| ∂Q
i
mal mode corresponding to a Jahn-Teller active mode, gives rise to off-diagonal matrix
elements in the Hamiltonian matrix. These non-zero off-diagonal matrix elements cause
the linear perturbation in V along the Jahn-Teller vibrational modes.
The non-Jahn-Teller modes do not couple directly with the Jahn-Teller modes. Thus
these modes can be treated as the usual harmonic oscillators to calculate the partition
functions. For a given Jahn-Teller mode only those off-diagonal elements are non-zero
that couple modes having the same Jahn-Teller quantum number j defined by j = l−Λ/2,
where Λ is the electronic quantum number and takes on values 1 or -1. The fact that
there is coupling between modes with the same j is used to convert the Hamiltonian into
a block diagonal form where each block corresponds to a specific value of j and is solved
independently.
Different Jahn-Teller modes couple with each other via a second order coupling in
which say a mode |Λ+ i|νi1 , li1 i|νi2 , li2 i is coupled to |Λ− i|νi1 +1, li1 −1i|νi2 , li2 i and also to
|Λ− i|νi1 , li1 i|νi2 + 1, li2 − 1i giving rise to coupling between the latter two modes. Figure 6
shows a schematic of how a given vibronic state containing two Jahn-Teller active modes
are coupled to five other vibronic states. Having this coupling greatly increases the size
and number of the blocks of the Hamiltonian, where each block now represents a pair of
Jahn-Teller quantum numbers j1 and j2 . Similarly having three active Jahn-Teller modes
further increases the size of the Hamiltonian blocks to be diagonalized. We have written
a small python code which can calculate the energy levels and subsequently entropy of
the Jahn-Teller active modes with up to two modes coupled with each other. This code is
later used to calculate the entropy of cyclopentadienyl (code available through supporting
materials).
To calculate the Hamiltonian matrix and then the energy levels we first need the
vibrational frequencies ωi and their contribution to the linear Jahn-Teller stabilization
energy Di for the E’2 Jahn-Teller vibrational modes. Kiefer et al 43 have calculated the
frequencies using finite differences of the analytical first derivative of CASSCF/cc-PVDZ
method at the conical intersection and found a negative mode of frequency -6345 cm−1
13
|±i|ν1 − 1, l1 ± 1i|ν2 , l2 i ↔
|±i|ν1 , l1 i|ν2 + 1, l2 ± 1i
l
|∓i|ν1 , l1 i|ν2 , l2 i
l
|±i|ν1 , l1 i|ν2 − 1, l2 ± 1i
↔ |±i|ν1 + 1, l1 ± 1i|ν2, l2 i
Figure 6: The figure shows that a given vibronic state |∓i|ν1 , l1 i|ν2 , l2 i having Jahn-Teller
quantum numbers j1 = l1 ± 0.5 and j2 = l2 ± 0.5 can couple directly with 5 other states
having the same values of j1 and j2 .
Mode
I
II
III
ωi
cm−1
815
1058
1411
Di
0.22
0.36
0.68
Table 4: Jahn-Teller active mode vibrational frequencies and values of Di ’s. 47 Di is the
contribution of each mode to Jahn-Teller stabilization. For example 815 cm−1 mode with
Di = 0.22 causes 815 × 0.22 = 179 cm−1 Jahn-Teller stabilization.
as the only active mode which gives rise to a 1655 cm−1 stabilization energy. Applegate
et al. 47,48 have performed detailed ab-initio calculations along with spectroscopic experiments to calculate the stabilization energy and active Jahn-Teller vibrational modes.
They have used GRHF 49 to calculate the vibrational frequencies of the unperturbed
molecule and then partitioned the Jahn-Teller stabilization energy between 3 of the 4 E’2
modes as shown in Table 4. Using these parameters they have got a good agreement with
experimentally observed spectrum. Due to the unusual nature of the potential energy
at the conical intersection it is difficult to judge whether either of the two methods used
gives the correct values of the Jahn-Teller frequencies.
In yet another approach Katzer and Sax 44 have developed a general procedure for
computing the moment of inertia for pseudo-rotations in Jahn-Teller active molecule.
According to this procedure the vibrational frequencies of the perturbed molecule are calculated and treated as harmonic oscillators and the low frequency vibration corresponding
to pseudo-rotation of the molecule is treated as a free rotor. At B3LYP/CBSB7 level of
theory the 2 B1 and 2 A2 structures with C2v symmetry are a first order saddle point and a
third order saddle point respectively. Instead we have performed B3LYP/CBSB7 calculation for cyclopentadienyl without using symmetry. The final optimized structure shows
one ultra low frequency mode of 25 cm−1 which corresponds to the pseudo-rotation. The
rotational constant to be used when calculating the partition function of the free rotor
for pseudo-rotation is taken from the work of Katzer and Sax as 230 cm−1 . The rest of
the modes are treated as harmonic oscillators.
In this paper we are mainly concerned about the thermochemistry of the cyclopentadienyl molecule and we use the different methods to calculate the entropy of cyclopentadienyl radical as a function of temperature. The results, plotted in Figure 7, suggest
14
that the difference between the various approaches for the Jahn-Teller effect is not very
large and is within the accuracy of the methods involved. For the calculation of partition
function and entropy SJT,M (see Figure 7) we have treated the three Jahn-Teller modes
independently of each other. These three modes are actually coupled to each other. The
coupling effect is important when one is looking at spectroscopic data, but it turns out
to be not important when one is interested in calculating macroscopic properties like
entropy. We have compared the entropy contribution of the two Jahn-Teller modes with
frequencies 815 cm−1 and 1058 cm−1 , with and without the coupling between the modes,
and found them to be equal to two decimal places. The purely rigid rotor harmonic oscillator approach using the B3LYP/CBSB7 frequencies results show a significant amount
of error when compared to the other methods. In the rest of the paper we have used
the Katzer and Sax approach to treat cyclopentadienyl because of its good accuracy and
relative computational simplicity.
Relative Entropy (cal/mole/K)
7
6
5
4
Sho
SJT,M
Sfr
3
2
1
0
-1
200
400
600
800 1000 1200 1400 1600 1800 2000
T (K)
Figure 7: c-C5 H5 entropy calculated using three different methods relative to the values
recommended by Kiefer et al. 43 which is represented by the dashed line at 0 cal/mole/K.
For Sho all modes of vibrations of a distorted molecule were treated as harmonic oscillators,
for Sf r the ultra low frequency of a distorted molecule is treated as free rotor and SJT,M
is the entropy calculated by diagonalizing the Hamiltonian using parameters suggested
by Applegate et al 47 (see text).
3.2
High Pressure Rate constant calculations
Figure 8 shows the optimized geometry of R2. A hydrogen atom can add to carbon 9
to form C5 H5 CH3 -5, to carbons 1 or 4 to form C5 H5 CH3 -1 or to carbons 2 or 3 to form
C5 H5 CH3 -2. For all the hydrogen atom additions the transitional modes consist of 2
angles which can take values to include the front and back side attack of H to R2. Atoms
15
H298
Molecule
kcal/mole
c-C5 H5
63.7
CH3
35.2
C5 H5 CH3 -5
27.1
C5 H5 CH3 -2
23.9
C5 H5 CH3 -1
24.2
Fulvene
52.7
R1
53.3
R2
53.3
R3
59.2
R4
77.2
S298
cal/mole-K
64.0
47.9
74.1
75.8
75.9
70.2
74.4
77.1
74.4
77.8
300
18.1
9.5
24.1
23.3
23.3
21.7
23.5
23.4
23.8
23.4
400
23.9
10.2
32.1
31.0
30.9
28.8
31.2
30.6
31.4
30.8
Cp (cal/mole-K)
500 600 800 1000 1500
28.6 32.3 37.6 41.3 46.9
10.9 11.6 12.9 14.1 16.3
38.8 44.3 52.3 58.0 66.7
37.7 43.2 51.5 57.5 66.4
37.6 43.1 51.5 57.4 66.4
34.6 39.3 46.1 50.8 57.9
37.5 42.5 49.9 55.1 62.9
36.7 41.6 48.9 54.1 61.9
37.7 42.7 50.0 55.2 62.9
36.9 41.8 49.0 54.1 61.8
Table 5: Thermochemical values calculated using CBS-QB3 level of theory. Special
methods are used for cyclopentadienyl, see Section 3.1.
Molecule
c-C5 H5 +CH3
C5 H5 CH3 -5
C5 H5 CH3 -2
C5 H5 CH3 -1
R1+H
R2+H
R4+H
H298 (kcal/mole)
CBS-QB3 BAC-MP4
98.9
98.8
27.1
25.8
23.9
23.7
24.2
23.7
105.4
104.8
105.4
106.5
129.3
132.1
Table 6: BAC-MP4 values are taken from paper by Melius et al. 2 The difference between
the values calculated by CBS-QB3 method and BAC-MP4 method in the worst case of
R4+H is less than 3 kcal/mole and in other cases is about 1 kcal/mole.
Figure 8: The optimized geometry of R2.
16
1 and 4 of R2 are identical to each other so the rate of attack of H atom on carbon 1 is
the same as the rate of attack of H atom on carbon 4. Thus the rate coefficient of attack
of H atom on carbon 1 is multiplied by 2 to get the total rate coefficient of H addition
on R2 to form C5 H5 CH3 -1. A similar approach is used to calculate the rate of attack of
H atom on R2 to form C5 H5 CH3 -2.
Using the approach described above, the rate of formation of C5 H5 CH3 -1 from R2 was
unexpectedly computed to be slower than the rate of formation of the other two isomers
by about an order of magnitude. Figure 9 shows the minimum energy pathway for each
C-H bond length calculated after randomly selecting 1000 configurations representing
different interfragment orientation and calculating the potential at each of those points.
It is striking that at a C-H bond distance of 2.9 Å the potential is very high for the
reaction forming C5 H5 CH3 -1. The transition state for formation of C5 H5 CH3 -1 at most
E and J is located at this interfragment distance. This seems highly suspicious and
we have performed an optimization of the R2+H structure at B3LYP/6-31G(d) level
of theory with the distance between carbon atom 1 and H atom frozen at 2.9 Å. This
geometry was then used to perform a calculation at CASPT2(6e,6o)/cc-pvtz level of
theory to give a barrier height of -15154 cm−1 which is very similar to the barrier heights
of the other two isomers at 2.9 Å. When we tried to perform a relaxation of the geometry
of the transition state for the other two isomers there was no drastic decrease in the
barrier. This shows that the high potential value is a consequence of the method we have
used, where by we fix the geometry of the fragments at their infinite separation distance.
To deal with this problem rigorously one would have to perform the calculation with
relaxation of the fragment taken into account. This would increase the complexity of the
problem significantly with the angular momentum expressions programmed in Variflex
for each of the set of internal coordinates τ in Equation 3 changing for each inter fragment
separation. This is beyond the scope of the present work. In this work we just remove
the point 2.9Å from our calculations for the C5 H5 CH3 -1 isomer which makes the rate of
formation of C5 H5 CH3 -1 from R2 very similar to the rate of formation of C5 H5 CH3 -2,
which is what one would expect.
For the addition of c-C5 H5 to CH3 the reactants have a symmetry number of 10 and 6
respectively. The transition state has a symmetry number of 12, which includes symmetry
number of 3 for rotation of CH3 about the C-C bonding axis, symmetry number of 2 for
the two dimensional rotational motions of CH3 about axes perpendicular to the C-C
bonding axis and symmetry number of 2 for rotation of c-C5 H5 about axes along the C-H
bond on the bonding carbon for c-C5 H5 . Thus the overall path degeneracy is 60/12=5 for
the five equivalent carbon atoms (equivalent due to the fast pseudo-rotation) in c-C5 H5 .
The rate coefficients of the barrierless reactions are shown in Figure 10. Also the rate
coefficients are fit to a modified Arrhenius form and are tabulated in Table 7.
3.3
Master chemical equation
While using the master equation to calculate the rate coefficients there are two scenarios
in which there are difficulties in calculating the phenomenological rate coefficients from
the chemically significant eigenvalues:
1. At low temperatures the lowest eigenvalue has a very small absolute value. Due
to the large difference between the lowest magnitude eigenvalue and higher magni17
0
-2000
Energy (cm-1)
-4000
-6000
-8000
-10000
-12000
-14000
-16000
C5H5CH3-2
C5H5CH3-5
C5H5CH3-1
-18000
-20000
2
2.5
3
3.5
4
4.5
5
C-H bond distance (A)
5.5
6
Figure 9: Minimum potential energy path for reaction of R2 with H to form C5 H5 CH3 -5,
C5 H5 CH3 -1 and C5 H5 CH3 -2.
Reactant
Product
C5 H5 CH3 -5
C5 H5 CH3 -1
c-C5 H5 +CH3
Fulvene+H2
Fulvene+H2
Fulvene+H2
R2+H
R2+H
R1+H
R2+H
R3+H
R4+H
C5 H5 CH3 -1
C5 H5 CH3 -2
C5 H5 CH3 -5
C5 H5 CH3 -5
C5 H5 CH3 -1
C5 H5 CH3 -2
C5 H5 CH3 -5
C5 H5 CH3 -1
C5 H5 CH3 -1
C5 H5 CH3 -2
C5 H5 CH3 -2
C5 H5 CH3 -5
k∞
A
2.8×1012
3.3×1013
1.1×10−10
1.2×10−7
1.6×10−12
1.7×10−13
3.8×10−10
5.4×10−10
1.1×10−10
3.0×10−10
1.8×10−10
2.2×10−10
n
1.2
2.1
-0.7
3.9
1.4
1.6
-0.1
0.3
0.6
0.1
0.5
0.6
Ea
24.8
25.1
-0.5
81.1
71.0
55.5
0.4
0.1
-0.2
0.0
-0.1
-0.8
Table 7: Computed high-pressure-limit rate coefficients given in the form k =
n
T
A 1000
exp(−Ea /R/T ). The unimolecular rate coefficients are in s−1 and the bimolecular rate coefficients are in cm3 /molecule/s. The units of Ea are kcal/mole.
tude eigenvalues, computers using 32-bit double precision numbers are not able to
calculate the eigenvalue correctly and a spurious positive eigenvalue is obtained.
2. At higher temperatures some of the chemically significant eigenvalues can merge
with eigenvalues associated with collisional relaxation. This condition usually sig-
18
1e-10
3
k (cm /molecule/s)
1e-09
1e-11
300
600
900 1200 1500 1800 2100
Temperature (K)
Figure 10: The high-pressure-limit rate coefficients of barrierless reactions. The list of
legends is:
+ is c-C5 H5 +CH3 → C5 H5 CH3 -5;
× is R2+H → C5 H5 CH3 -5;
∗ is R2 + H → C5 H5 CH3 -1;
is R1 + H → C5 H5 CH3 -1;
is R2 + H → C5 H5 CH3 -2;
◦ is R3 + H → C5 H5 CH3 -2;
• is R4 + H → C5 H5 CH3 -5;
nifies that two species are rapidly equilibrated and essentially act as one chemical
species. In other words the rate of conversion between the two species is equal to
or faster than the rate of relaxation of internal modes in the two species.
We have plotted the first 10 eigenvalues as a function of temperature at 1 atm pressure
in Figure 11. It shows that at almost all temperatures we see that two chemically
significant eigenvalues are very close to eigenvalues describing relaxation of internal mode.
These two fastest chemically significant eigenvalues correspond to the equilibria between
the isomers of the C6 H8 adduct. The slowest chemically significant eigenvalue corresponds
to the formation of the products from the wells and the reactants and the remaining
slow eigenvalue corresponds to the equilibrium between the bimolecular reactants and
well species. In the present system we treat the three rapidly equilibrated well species
C5 H5 CH3 -5, C5 H5 CH3 -2 and C5 H5 CH3 -3 as a single species C5 H5 CH3 . To calculate the
density of states and partition function of this pseudo species one would ideally pick out a
vibrational mode that is closest to the intramolecular hydrogen transfer which results in
the isomerization of the three isomers. The density of states of this vibrational mode will
be replaced by states calculated by solving the Schroedinger equation numerically for the
potential energy surface formed by the intramolecular migration of hydrogen atom. This
19
procedure is analogous to the one usually employed to correct for the internal hindered
rotors. But in this paper we have forgone the full procedure in interest of simplicity. We
here take one of the isomers and multiply its density of states and partition function by 3
to get the density of states and partition functions of the pseudo species. Because the three
isomeric species have very similar ground state energies and vibrational and rotational
states the sum of the density of states of the three isomers is very well approximated by
multiplying one of the species’ density of states by 3. We have confirmed this by carrying
out calculations by taking each of these isomers separately and using each of their density
of states to calculate the density of state of the pseudo species and the final results for
rate coefficients change very little.
With our pseudo species we reformulate the master chemical equation and obtain 2
chemically significant eigenvalues which are separated from the continuum of eigenvalues.
These two eigenvalues can be used to calculate the rate coefficients of interest.
Below 900 K the slowest eigenvalue has a very small magnitude and our eigenvalue
solver tends to give a spurious positive value. In this work we have focused primarily on
temperatures above 900 K where we can extract accurate rate coefficients.
1e+08
eigenvalues (1/s)
1e+06
10000
100
1
0.01
0.0001
800
1000
1200 1400 1600
Temperature (K)
1800
2000
Figure 11: The values of first 10 slowest eigenvalues calculated at various temperatures at
1 atm pressure. Above 1300 K, the fastest two chemically significant eigenvalues become
indistinguishable from the rest of the eigenvalues.
3.4
Rate coefficients for c-C5H5 +CH3 → C5 H5 CH3 and products
We have calculated the rate of formation of the equilibrated C5 H5 CH3 and bimolecular
products from c-C5 H5 +CH3 and have plotted the rate coefficient in Figure 12 for pressure
of 1 atm and 0.01 atm N2 . The figures show a more pronounced fall off at lower pressure
along with a higher rate of direct conversion of reactant to the products.
20
3
k (cm /molecule/s)
1e-09
1e-10
1e-11
1e-12
1e-13
1e-14
1e-15
1e-16
1e-17
1e-18
1e-19
800
1200
1600
Temperature (K)
2000
3
k (cm /molecule/s)
(a) p=0.01 atm
1e-09
1e-10
1e-11
1e-12
1e-13
1e-14
1e-15
1e-16
1e-17
1e-18
1e-19
800
1200
1600
Temperature (K)
2000
(b) p=1 atm
Figure 12: The rate coefficient of formation of the C5 H5 CH3 species and various bimolecular products from c-C5 H5 +CH3 at pressure of 0.01 atm and 1 atm N2 . The solid line represents high-pressure-limit rate coefficient for recombination of c-C5 H5 +CH3 , the dashed
line represents the actual rate coefficient for reaction c-C5 H5 +CH3 → C5 H5 CH3 . The rest
of the lines represent rate coefficients to products where, + corresponds to Fulvene+H2 ;
× corresponds to R2+H; ∗ corresponds to R1+H; corresponds to R3+H; corresponds
to R4+H2
21
Note on Detailed Balance
There are two approximations that are commonly made to calculate the rotational partition function of an asymmetric top because calculating its exact quantum states is
computationally demanding. The first one involves treating the three rotations independently and then multiplying the classical partition function of each of them to calculate
the total rotational partition function as shown below.
Qrot
π 1/2
=
σ
8π 2 IA kB T 8π 2 IB kB T 8π 2 IC kB T
h2
h2
h2
1/2
In the second approximation the molecule is treated as a symmetric top even if it is an
asymmetric top by calculating the moment of inertia about the degenerate axes IA as the
geometric mean of the moment of inertia about the nearly degenerate axes. With this
assumption the rotational partition function Qrot can be calculated as
Qrot =
JX
max K=J
X
J=0
J(J + 1)A + (B − A)K 2
(2J + 1) exp −
kB T
K=−J
where A and B are the rotational constants of the molecule about the degenerate and
unique axis respectively and Jmax is a sufficiently large value. As mentioned in Section 2.2.1 to calculate N(E, J) of a molecule Variflex treats it as a symmetric top. The
partition function calculated by the two approximations can differ from each other by
about 10%. Thus if the partition function is calculated using the first approximation and
as mentioned in Section 2.2.1 the second approximation is used to calculate the density
of states used in the master equation the detailed balance is not satisfied. This shows
the importance of calculating the partition function and density of states in a consistent
manner.
In Figure 13 we plot the rate coefficient for the decomposition of 5-methylcyclopentadiene
in Kr measured in a shock tube by Ikeda et al. 40 against the rate coefficient predicted
by our calculations for a pressure of 200 torr. Also on the same graph is plotted the
decomposition rate of methylcyclopentadiene in Ar measured in a shock tube at pressures of 2.71 atm by Lifshitz et al. 50 The calculated rate coefficients are systematically
higher by about a factor of 2 for all the temperatures measured. This factor of 2 is well
within the error bounds of the calculations. For the comparison with data by Lifshitz we
have performed calculations using two different energy transfer parameters for Ar gas.
When we use the value of < ∆Ed >= 150 × (T /300) cm−1 , which has been recommended
by Golden et al. 51 we get about a fact of 5 difference between calculated results and
experimental measurements, but when we change this value to < ∆Ed >= 100 cm−1
the agreement becomes much better. < ∆Ed > values have large uncertainties and it is
difficult to pin down an exact value a priori. In this study we have chosen to use the
value of < ∆Ed >= 100 for Ar to carry out the rest of the calculations because this value
gives good agreement with the experimental results. We should note that the error in
the calculated values of the rate coefficients can be as high at a factor of 5 as is shown in
Section 3.5.
22
1e+07
1e+06
k (1/s)
1e+05
1e+04
1e+03
1e+02
1e+01
1e+00
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
-1
1000/T (K )
Figure 13: The ’+’ signs are measurements of C5 H5 CH3 -5 decomposition by Ikeda et al. 40
at 100-400 torr Kr and the solid line are calculations performed at 200 torr Kr using <
∆Ed >= 350 cm−1 . The solid boxes are measurements by Lifshitz et al. 50 at 2.71 atm Ar;
the dotted line is from calculations at the same pressure using < ∆Ed >= 150 × (T /300)
cm−1 and the dashed line is using < ∆Ed >= 100 cm−1 .
23
3.5
Rate coefficients of formation of bimolecular products from
C5 H5 CH3
The rate of formation of bimolecular products is one of the most important results of
this study. The C6 H7 products further lose a hydrogen atom and then isomerize to
form benzene. In literature many pathways have been mentioned which lead to benzene
formation from C6 H7 . 2,52 In some practical situations these C6 H7 molecules are formed
predominantly by the pathways in Figure 1. At most temperatures and pressures the
formation of R2 is about an order of magnitude greater than the the second fastest
forming product R1+H (see Figure 14). But R2 does not have a direct route that leads
to the formation of benzene without going through one of the other C6 H7 radicals. To
calculate the formation of benzene from these C6 H7 radicals a similar analysis as the one
presented in this paper needs to be performed.
1e+00
branching ratio
1e-01
1e-02
1e-03
1e-04
1e-05
1e-06
1e-07
1e-08
800
1000 1200 1400 1600 1800 2000
T (K)
Figure 14: The branching ratio for each product channel from decomposition of C5 H5 CH3
at 1 atm N2 . + corresponds to Fulvene+H2 ; × corresponds to R2+H; ∗ corresponds to
R1+H; corresponds to R3+H; corresponds to R4+H2
24
Reactions
900
1100
−11
25
c-C5 H5 +CH3 →C5 H5 CH3
c-C5 H5 +CH3 →Fulvene+H2
c-C5 H5 +CH3 →R2+H
c-C5 H5 +CH3 →R1+H
c-C5 H5 +CH3 →R3+H
c-C5 H5 +CH3 →R4+H
C5 H5 CH3 →c-C5 H5 +CH3
C5 H5 CH3 →Fulvene+H2
C5 H5 CH3 →R2+H
C5 H5 CH3 →R1+H
C5 H5 CH3 →R3+H
C5 H5 CH3 →R4+H
5.9×10
3.3×10−18
9.1×10−13
2.5×10−14
1.8×10−15
6.8×10−19
9.3×10−2
2.3×10−10
1.1×10−4
2.5×10−6
3.5×10−8
—a
c-C5 H5 +CH3 →C5 H5 CH3
c-C5 H5 +CH3 →Fulvene+H2
c-C5 H5 +CH3 →R2+H
c-C5 H5 +CH3 →R1+H
c-C5 H5 +CH3 →R3+H
c-C5 H5 +CH3 →R4+H
C5 H5 CH3 →c-C5 H5 +CH3
C5 H5 CH3 →Fulvene+H2
C5 H5 CH3 →R2+H
C5 H5 CH3 →R1+H
C5 H5 CH3 →R3+H
C5 H5 CH3 →R4+H
1.4×10−10
1.6×10−18
3.6×10−13
1.1×10−14
1.1×10−15
6.6×10−19
2.2×10−1
3.0×10−9
9.9×10−4
2.6×10−5
1.2×10−6
3.2×10−11
Temperatures (K)
1300
1500
p=0.01 atm
1.2×10−11 1.8×10−12
1.2×10−17 3.2×10−17
2.4×10−12 4.7×10−12
7.5×10−14 1.6×10−13
8.8×10−15 2.8×10−14
1.9×10−17 2.0×10−16
2.3×101
4.4×102
7.7×10−8 1.7×10−6
3.6×10−2 8.0×10−1
8.1×10−4 1.8×10−2
1.2×10−5 2.4×10−4
—a
—a
p=1 atm
7.3×10−11 2.5×10−11
1.0×10−17 3.1×10−17
1.8×10−12 4.4×10−12
5.9×10−14 1.5×10−13
7.8×10−15 2.7×10−14
1.9×10−17 2.0×10−16
1.4×102
6.1×103
3.9×10−6 2.5×10−4
1.1×100
6.9×101
3.1×10−2
1.9×100
1.9×10−3 1.3×10−1
9.2×10−8 8.4×10−6
1700
2000
2.3×10−13
7.3×10−17
7.8×10−12
2.9×10−13
6.5×10−14
1.1×10−15
2.4×103
9.3×10−6
4.6×100
1.0×10−1
1.1×10−3
—a
3.1×10−14
1.4×10−16
1.1×10−11
4.6×10−13
1.3×10−13
3.8×10−15
7.0×103
2.8×10−5
1.4×101
3.2×10−1
2.4×10−3
—a
1.8×10−15
2.7×10−16
1.6×10−11
6.8×10−13
2.3×10−13
1.1×10−14
2.2×104
8.5×10−5
4.5×101
1.0×100
5.5×10−3
—a
6.5×10−12
7.2×10−17
7.7×10−12
2.9×10−13
6.5×10−14
1.1×10−15
5.5×104
2.7×10−3
7.3×102
2.1×101
1.5×100
1.1×10−4
1.4×10−12
1.4×10−16
1.1×10−11
4.5×10−13
1.3×10−13
3.8×10−15
2.2×105
1.2×10−2
3.1×103
9.0×101
6.7×100
5.1×10−4
1.4×10−13
2.7×10−16
1.6×10−11
6.8×10−13
2.3×10−13
1.1×10−14
8.3×105
5.0×10−2
1.3×104
3.8×102
2.9×101
2.2×10−3
Table 8: Rate coefficients for various reactions for temperature range 900-2000 K at 0.01 atm and 1 atom pressure N2 . In this table
C5 H5 CH3 represents equilibrated isomers. The unimolecular rate coefficients are in s−1 and the bimolecular rate coefficients are in
cm3 /molecule/s. a Too slow to be computed accurately using eigenvalue approach at double precision.
Reactions
5 H5 +CH3 →C5 H5 CH3
900
1100
−10
26
c-C5 H5 +CH3 →Fulvene+H2
c-C5 H5 +CH3 →R2+H
c-C5 H5 +CH3 →R1+H
c-C5 H5 +CH3 →R3+H
c-C5 H5 +CH3 →R4+H
C5 H5 CH3 →c-C5 H5 +CH3
C5 H5 CH3 →Fulvene+H2
C5 H5 CH3 →R2+H
C5 H5 CH3 →R1+H
C5 H5 CH3 →R3+H
C5 H5 CH3 →R4+H
1.5×10
5.0×10−19
9.9×10−14
3.1×10−15
3.8×10−16
4.8×10−19
2.4×10−1
4.7×10−9
1.4×10−3
3.8×10−5
2.3×10−6
3.4×10−10
c-C5 H5 +CH3 →C5 H5 CH3
c-C5 H5 +CH3 →Fulvene+H2
c-C5 H5 +CH3 →R2+H
c-C5 H5 +CH3 →R1+H
c-C5 H5 +CH3 →R3+H
c-C5 H5 +CH3 →R4+H
C5 H5 CH3 →c-C5 H5 +CH3
C5 H5 CH3 →Fulvene+H2
C5 H5 CH3 →R2+H
C5 H5 CH3 →R1+H
C5 H5 CH3 →R3+H
C5 H5 CH3 →R4+H
1.6×10−10
8.6×10−20
1.5×10−14
5.0×10−16
6.9×10−17
1.6×10−19
2.5×10−1
5.4×10−9
1.5×10−3
4.2×10−5
2.8×10−6
8.8×10−10
Temperatures (K)
1300
1500
p=10 atm
1.1×10−10 5.8×10−11
6.1×10−18 2.7×10−17
9.8×10−13 3.5×10−12
3.3×10−14 1.3×10−13
5.1×10−15 2.4×10−14
1.8×10−17 2.0×10−16
2.0×102
1.4×104
1.2×10−5 1.4×10−3
2.8×100
3.0×102
8.2×10−2
9.2×100
7.2×10−3 9.5×10−1
2.7×10−6 5.5×10−4
p=100 atm
1.2×10−10 9.1×10−11
2.1×10−18 1.6×10−17
2.8×10−13 1.8×10−12
1.0×10−14 6.7×10−14
1.8×10−15 1.5×10−14
1.1×10−17 1.7×10−16
2.3×102
2.2×104
2.0×10−5 4.2×10−3
4.2×100
7.4×102
1.3×10−1
2.4×101
1.4×10−2
3.4×100
−5
1.7×10
8.0×10−3
1700
2000
2.3×10−11
7.0×10−17
7.2×10−12
2.7×10−13
6.4×10−14
1.1×10−15
1.8×105
2.5×10−2
5.1×103
1.6×102
1.8×101
1.3×10−2
7.1×10−12
1.4×10−16
1.1×10−11
4.5×10−13
1.2×10−13
3.8×10−15
9.2×105
1.5×10−1
2.9×104
9.1×102
1.1×102
9.1×10−2
1.0×10−12
2.7×10−16
1.6×10−11
6.8×10−13
2.3×10−13
1.1×10−14
4.2×106
7.7×10−1
1.5×105
4.7×103
5.9×102
5.5×10−1
5.4×10−11
5.8×10−17
5.4×10−12
2.1×10−13
5.3×10−14
1.1×10−15
4.3×105
1.3×10−1
2.1×104
7.1×102
1.1×102
3.8×10−1
2.5×10−11
1.3×10−16
1.0×10−11
4.1×10−13
1.2×10−13
3.7×10−15
2.9×106
1.2×100
1.7×105
6.1×103
1.0×103
4.3×100
5.8×10−12
2.7×10−16
1.5×10−11
6.7×10−13
2.2×10−13
1.1×10−14
1.7×107
8.8×100
1.2×106
4.3×104
7.8×103
3.9×101
Table 9: Rate coefficients for various reactions for temperature range 900-2000 K at 10 atm and 100 atm pressure N2 . In this table
C5 H5 CH3 represents equilibrated isomers. The unimolecular rate coefficients are in s−1 and the bimolecular rate coefficients are in
cm3 /molecule/s.
The rate coefficients for all the pressure dependent reactions at various pressures and
temperatures in N2 bath gas are given in Table 8 and Table 9. Rate coefficients for all
these reactions in Ar bath gas are included in supplementary information.
Uncertainties in calculated rates
The rates calculated in Table 8 and Table 9 are based on a long series of calculation
steps each of which have uncertainties and approximations associated with them. The
cumulative effect of all these uncertainties are difficult to quantify, but we feel that it is
useful to have an estimate on the uncertainties of the rate coefficients. For most of the
calculations performed we have used the CBS-QB3 compound method to calculate the
energies of molecules and saddle points. This method is expected to typically have an
error of about 2 kcal/mole. Based on this we assume that the error in the barrier heights
an be about 2 kcal/mole. The density of states and frequencies are usually calculated
more accurately than the energy, so we expect the error in entropies to be less than
1.5 cal/mole/K. A 2 kcal/mole error in enthalpy and 1.5 cal/mole/K error in entropy
translates to an uncertainty in the computed rate coefficients of about a factor of 5 at
1000 K.
4
Conclusion
In this paper we have calculated the high-pressure-limit and pressure dependent rate coefficients for all the reactions in Figure 1. We have compared our calculations against shock
tube experiments and the comparison is quite good. We see a significant amount of fall off
and chemical activation in the pressure dependent network and using high pressure rate
coefficients for these reactions would not be appropriate. The rate coefficients computed
here will be helpful in understanding the formation of aromatic rings in combustion.
5
Acknowledgement
We would like to acknowledge Dr. Stephen Klippenstein for giving us a newer version of
Variflex prior to its official release and for helping us get started with it. This work was
supported by the US Department of Energy through grant DEGC04-01AL67611.
6
Supporting Information
Python code to calculate the energy levels and entropy for Jahn-Teller active mode is
included. Cartesian coordinates, harmonic vibrational frequencies and principal moments
of inertia of all the molecules and transition states are given. Also rate coefficients of all
the reactions at 0.01, 1, 10 and 100 atm Ar are given. This material is available free of
charge via the Internet at http://pubs.acs.org.
27
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